Right now ridable animals, horse/donky/llama regen health at a slow rate, and tamed wolves and horse/donkey/mule can be regened by feeding them. Would be nice if one of those, feeding i would go with, was extended to all tamable and breedable pets/friendly mobs, as well as for when they eat for themselves.

Right now keeping individual animals alive is pretty high effort, and mid to late game for most players i would guess. Its quite stressfull to constantly be keeping them away from even small amount of falldamage and stuff that might acumuulate without you notesing for the entire time you have your world running, so they just end up being kept indore or in pens... Yes, it could be an insentive to engage in more of the mechanics of the game for it to be more exiting, but for people who just want to build pretty houses and stuff its just annoying, and for people who want to do fansy stuff and play lots of the endgame suff there already is one, to heal yourselfe... some people want more cosy than exiting...

Would make gathering animals from far away more feasable and not so grindy and lategame, even then, using potions is mostly an extreme emergensy usecase for more casual players, not something you carry around stacks of in shulker boxes to casualy make certain your pet has full health...

Not a good idea. The reason is that if a player is fighting you the game will become obviously less challenging and way much easier than before. Remember that this is reality too so in the game it makes sense to do injure your own pet. Just practice not to hit them